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In an important paper, Weinstein and Yildiz (2007) show that if players have an infinite depth of reasoning and this is commonly believed, types generically have a unique rationalizable action in games that satisfy a richness condition. We show that this result does not extend to environments where players may have a finite depth of reasoning, or think it is possible that the other player has a finite depth of reasoning, or think that the other player may think that is possible, and so on, even if this so-called "grain of naivete" is arbitrarily small. More precisely, we show that even if there is almost common belief in the event that players have an infinite depth of reasoning, there are types with multiple rationalizable actions, and the same is true for "nearby" types. Our results demonstrate that both uniqueness and multiplicity are robust phenomena when we relax the assumption that it is common belief that players have an infinite depth, if only slightly.

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